Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 83
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The location of thismaximum value is found to shift to higher η as Pr is decreased. The velocity boundaryLines: 352 to 396layer thickness is also found to increase as Pr is decreased to low values. These trends———are expected from the physical mechanisms that govern this boundary layer flow, as2.09578ptPgVardicussed earlier. It is also worth noting that the results indicate the coupling between———the velocity and temperature fields, as evidenced by the presence of flow whereverCustom Page (1.0pt)a temperature difference exists, such as the profiles at low Pr.
Additional results anddiscussion on the flow are given in several books; see, for instance, the books by * PgEnds: EjectKaviany (1994), Bejan (1995), and Oosthuizen and Naylor (1999).The heat transfer from the heated surface may be obtained as[535], (11)1/4 ∂T∂θ1 Gr x= −k(Tw − T∞ )qx = −k∂y 0x4∂η 0 k(Tw − T∞ ) Gr x 1/4= −θ (0)(7.30)x4The local Nusselt number Nux is given byNux =xhx xqx=kTw − T ∞ kWe have for an isothermal surface Gr x 1/4−θ (0)1/4= √ Gr 1/4(7.31)Nux = −θ (0)x = φ(Pr)Gr x42 √where φ(Pr) = −θ (0) / 2. Therefore, the local surface heat transfer coefficienthx varies ask[−θ (0)] gβ(Tw − T∞ ) 1/4where B =hx = Bx −1/4√ν22BOOKCOMP, Inc.
— John Wiley & Sons / Page 535 / 2nd Proofs / Heat Transfer Handbook / Bejan5360.3f ⬘()0.2Pr =0.010.10681012141618202224ux2ν公Grx0.7f ⬘() =123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION0.6[536], (12)Pr =0.01Lines: 396 to 402———*0.5———Normal Page* PgEnds: PageBreak0.40.30.258.25099pt PgVar[536], (12)0.72120.11001001000123( (y Gr x= x441/4567Figure 7.3 Calculated velocity distributions in the boundary layer for flow over an isothermalvertical surface. (From Ostrach, 1953.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 536 / 2nd Proofs / Heat Transfer Handbook / BejanLAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES0.6 ()0.4Pr =0.010.201.0681012141618202224[537], (13)Pr = 0.010.8Lines: 402 to 419T ⫺ T⬁Tw ⫺ T⬁———() =1234567891011121314151617181920212223242526272829303132333435363738394041424344455370.71503pt PgVar———Normal Page* PgEnds: Eject0.60.720.4[537], (13)10.210010100002123y Gr x= x4( (1/4456Figure 7.4 Calculated temperature distributions in the boundary layer for flow over anisothermal vertical surface. (From Ostrach, 1953.)The average value of the heat transfer coefficient h̄ may be obtained by averagingthe heat transfer over the entire length of the vertical surface, to yieldh̄ =1LLhx dx =0Therefore,BOOKCOMP, Inc.
— John Wiley & Sons / Page 537 / 2nd Proofs / Heat Transfer Handbook / Bejan4 B 3/4L3L538123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION4 −θ (0)44Gr 1/4 = φ(Pr) Gr 1/4 = NuLNu =√3332(7.32)The values of φ(Pr) can be obtained from a numerical solution of the governingdifferential equations. Values obtained at various Pr are listed in Table 7.1. Thesignificance of n and the uniform heat flux data in the table is discussed later. Anapproximate curve fit to the numerical results for φ(Pr) has been given by Oosthuizenand Naylor (1999) asφ(Pr) =0.316Pr 5/42.44 + 4.88Pr 1/2 + 4.95Pr1/4(7.33)It must be mentioned that these results can be used for both heated and cooled surfaces(i.e., Tw > or < T∞ ), yielding respectively a positive q value for heat transfer fromthe surface and a negative value for heat transfer to the surface.In several problems of practical interest, the surface from which heat transferoccurs is nonisothermal.
The two families of surface temperature variation that giverise to similarity in the governing laminar boundary layer equations have been shownby Sparrow and Gregg (1958) to be the power law and exponential distributions,given asTw − T∞ = Nx nandTw − T∞ = MemxTABLE 7.1 Computed Values of the Parameter φ(Pr) for a VerticalHeated Surfaceφ(Pr)φ Pr, 15(Isothermal),(Uniform Heat Flux),Prn=0n = 1500.010.720.7331.02.02.55.06.77.010102103104∞0.600Pr1/20.05700.357—0.4010.507—0.675—0.7540.8261.552.805.010.503Pr1/4Source: Gebhart (1973).BOOKCOMP, Inc.
— John Wiley & Sons / Page 538 / 2nd Proofs / Heat Transfer Handbook / Bejan0.711Pr1/20.0669—0.410——0.616—0.829—0.9311.74——0.563Pr1/4(7.34)[538], (14)Lines: 419 to 472———1.31822pt PgVar———Normal PagePgEnds: TEX[538], (14)LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445539where N, M, n, and m are constants. The power law distribution is of particularinterest, since it represents many practical circumstances.
The isothermal surface isobtained for n = 0. From the expression for qx , eq. (7.30), it can be shown that qxvaries with x as x (5n−1)/4 . Therefore, a uniform heat flux condition, qx = constant,arises for n = 15 . It can also be shown that physically realistic solutions are obtainedfor − 35 ≤ n < 1 (Sparrow and Gregg, 1958; Jaluria, 1980). The governing equationsare obtained for the power law case asf + (n + 3)ff − 2(n + 1)(f )2 + θ = 0(7.35)θ+ (n + 3)f θ − 4nf θ = 0Pr(7.36)The local Nusselt number Nux is obtained as[539], (15)Nux−θ (0)== φ(Pr, n)√Gr 1/42xNux /Gr 1/4x(7.37)Lines: 472 to 517− 35 ,is plotted against n in Fig. 7.5. For n <the functionThe functionis found to be negative, indicating the physically unrealistic circumstance of heattransfer to the surface for Tw > T∞ .
The surface is adiabatic for n = − 35 , which thusrepresents the case of a line source at the leading edge of a vertical adiabatic surface,so that no energy transfer occurs at the surface for x > 0.For the case of uniform heat flux, n = 15 and qx = q , a constant. Therefore, fromeq. (7.30), gβN 1/4q = k −θ (0) N4ν2which givesN=qk[−θ (0)]4/5 4ν2gβ1/5(7.38)Therefore, for a given heat flux q , which may be known, for example, from theelectrical input into the surface, the temperature of the surface varies as x 1/5 andits magnitude may be determined as a function of the heat flux and fluid propertiesfrom eq. (7.38). The parameter −θ (0) is obtained from a numerical solution of thegoverning equations for n = 15 at the given value of Pr.
Some results obtained fromGebhart (1973) are shown in Table 7.1 as φ Pr, 15 .7.3.2Inclined and Horizontal SurfacesIn many natural convection flows, the thermal input occurs at a surface that is itselfcurved or inclined with respect to the direction of the gravity field.
Consider, first, aflat surface at a small inclination γ from the vertical. Boundary layer approximations,BOOKCOMP, Inc. — John Wiley & Sons / Page 539 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.53522pt PgVar———Normal PagePgEnds: TEX[539], (15)5401.21.00.80.60.4Nux /(Grx /4)1/4123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONPr = 1.0[540], (16)Pr = 0.70.2Lines: 517 to 522⫺1.0 ⫺0.50———00.51.0⫺0.21.52.02.53.0n⫺0.4⫺0.6⫺0.8⫺1.0Figure 7.5 Dependence of the local Nusselt number on the value of n for a power law surfacetemperature distribution.
(From Sparrow and Gregg, 1958.)similar to those for a vertical surface, may be made for this flow. It can be shown thatif x is taken along the surface and y normal to it, the continuity and energy equations,eqs. (7.15) and (7.17), respectively, remain unchanged and the x-direction momentumequation becomes∂u1 ∂∂u∂uu+v= gβ(T − T∞ ) cos γ +µ(7.39)∂x∂yρ ∂y∂yBOOKCOMP, Inc.
— John Wiley & Sons / Page 540 / 2nd Proofs / Heat Transfer Handbook / Bejan0.09702pt PgVar———Normal Page* PgEnds: Eject[540], (16)LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445541Therefore, the problem is identical to that for flow over a vertical surface except that gis replaced by g cos γ in the buoyancy term. Therefore, a replacement of g by g cos γin all the expressions derived earlier for a vertical surface would yield the corresponding results for an inclined surface.
This implies using Gr x cos γ for Gr x and assumingequal rates of heat transfer on the two sides of the surface. This is strictly not the casesince the buoyancy force is directed away from the surface at the top and toward thesurface at the bottom, resulting in differences in boundary layer thicknesses and heattransfer rates. However, this difference is neglected in this approximation.The preceding procedure for obtaining the heat transfer rate from an inclinedsurface was first suggested theoretically by Rich (1953), and his data are in goodagreement with the values predicted.
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